Determine if Continuous M(x)=(x-1)/(9x^2-16)

Solution:

Step:1

Identify the domain to check the continuity of the function.

Step:1.1

To find the points of discontinuity, equate the denominator of $M(x)=\frac{x-1}{9x^2-16}$ to zero: $9x^2-16=0$.

Step:1.2

Determine the values of $x$ that satisfy the equation.

Step:1.2.1

Isolate $9x^2$ by adding $16$ to both sides: $9x^2=16$.

Step:1.2.2

Divide the equation $9x^2=16$ by $9$ to simplify it.

Step:1.2.2.1

Perform the division: $\frac{9x^2}{9}=\frac{16}{9}$.

Step:1.2.2.2

Reduce the left-hand side of the equation.

Step:1.2.2.2.1

Eliminate the common factor of $9$: $\frac{\cancel{9}{x}^2}{\cancel{9}}=\frac{16}{9}$.

Step:1.2.2.2.1.2

Simplify to $x^2=\frac{16}{9}$.

Step:1.2.3

Extract the square root of both sides to solve for $x$: $x=\pm \sqrt{\frac{16}{9}}$.

Step:1.2.4

Simplify the square root expression $\pm \sqrt{\frac{16}{9}}$.

Step:1.2.4.1

Express the square root as a fraction: $x=\pm \frac{\sqrt{16}}{\sqrt{9}}$.

Step:1.2.4.2

Simplify the numerator.

Step:1.2.4.2.1

Express $16$ as $4^2$: $x=\pm \frac{\sqrt{4^2}}{\sqrt{9}}$.

Step:1.2.4.2.2

Extract terms from under the radical: $x=\pm \frac{4}{\sqrt{9}}$.

Step:1.2.4.3

Simplify the denominator.

Step:1.2.4.3.1

Express $9$ as $3^2$: $x=\pm \frac{4}{\sqrt{3^2}}$.

Step:1.2.4.3.2

Extract terms from under the radical: $x=\pm \frac{4}{3}$.

Step:1.2.5

Combine the positive and negative solutions to find the complete solution.

Step:1.2.5.1

Use the positive value from $\pm$: $x=\frac{4}{3}$.

Step:1.2.5.2

Use the negative value from $\pm$: $x=-\frac{4}{3}$.

Step:1.2.5.3

The complete solution includes both positive and negative values: $x=\frac{4}{3},-\frac{4}{3}$.

Step:1.3

The domain consists of all $x$ values that do not make the expression undefined. In interval notation: $(-\mathrm{\infty },-\frac{4}{3})\cup (-\frac{4}{3},\frac{4}{3})\cup (\frac{4}{3},\mathrm{\infty })$. In set-builder notation: $\{x|x\ne \frac{4}{3},-\frac{4}{3}\}$.

Step:2

Given that the domain excludes certain real numbers, the function $M(x)=\frac{x-1}{9x^2-16}$ is not continuous for all real numbers.

Step:3

Knowledge Notes:

1. Domain of a Function: The set of all possible input values (usually 'x') for which the function is defined. A function is continuous if it is defined and uninterrupted for all values in its domain.

2. Continuity: A function is continuous at a point if the limit as it approaches the point from both sides equals the function's value at that point. A function is continuous over an interval if it is continuous at every point in that interval.

3. Discontinuity: A point where a function is not continuous. Discontinuities can occur where the function is not defined (such as division by zero) or where the limit does not exist.

4. Solving Quadratic Equations: A quadratic equation in the form $a{x}^2+bx+c=0$ can be solved by factoring, completing the square, or using the quadratic formula. In this case, the equation is a simple binomial that can be factored as a difference of squares.

5. Difference of Squares: A binomial in the form $a^2-b^2$ can be factored into $(a+b)(a-b)$.

6. Interval Notation: A mathematical notation used to represent an interval on the real number line. It uses parentheses '()' for open intervals and brackets '[]' for closed intervals.

7. Set-Builder Notation: A notation used to describe a set by stating the properties that its members must satisfy. For example, $\{x|x\ne a\}$ describes a set of all elements 'x' such that 'x' is not equal to 'a'.

8. Square Roots: The square root of a number 'a' is a value 'b' such that $b^2=a$. The principal square root is the non-negative root and is denoted by $\sqrt{a}$.

9. Rationalizing the Denominator: The process of eliminating radicals from the denominator of a fraction by multiplying the numerator and the denominator by an appropriate value that will make the denominator rational.